If you send a bitcoin payment to a seller, do they see your IP address?

Since you send bitcoin payments through one of the Bitcoin clients, I assume it announces the payment to the network, so each computer it connects to in order to do so would know the IP address from which you announce the payment.

So when you make a bitcoin payment, who gets your address IP, and also which of them will know you are the originator of the transaction message and thus the wallet owner, and not just re-broadcasting it (which I assume is what the network does)?

Does the sellers themselves see your IP address, and can they be reasonably sure that they are seeing the IP address of the wallet's owner (and not somebody else)?